32
2 Preliminaries
The families of dissipation potentials π( ˙
α; γ ) and dual dissipation potentials π
∗
(a; γ )
are depicted together with their sub-differentials a = a( ˙
α; γ ) and ˙
α = ˙
α(a; γ ) in
Figs. 2.2 and 2.3.
−1
−0.5
0
0.5
1
0.2
0
0.2
a
π
∗
1 (a; γ)
γ
1 + γ
|a|
1+γ
γ
γ
∞
∞
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
a
˙
α 1 (a; γ)
γ
−1
−0.5
0
0.5
1
0.2
0
0.2
a
π
∗
2 (a; γ)
γ [1 −
√
1 − a 2 ]
γ
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
a
˙
α 2 (a; γ)
γ
−1
−0.5
0
0.5
1
0.2
0
0.2
a
π
∗
3 (a; γ)
γ [a artanh a + ln
√
1 − a 2 ]
γ
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
a
˙
α 3 (a; γ)
γ
Fig. 2.3 Alternative families of dual dissipation potentials π ∗ (a; γ ) together with resulting
sub-differentials ˙
α = ˙
α(a; γ ) for regularization parameter γ ∈ {0, 0.1, 0.2, 0.3}. For γ → 0 all
π ∗ (a; γ ) converge to the non-smooth indicator function I {|a|−1} of |a| ≤ 1